Clean and collect a week's worth of items that you'd normally throw away and challenge your children to think of ways you could use the items in crafting.
The following are suggestions to get you started thinking in the Trashy Crafts mode. I've given brief descriptions and, in some cases, illustrations to make the idea more clear. Keep your eyes open for ideas all around you. Look at craft fairs at what other people are doing with castaways. Magazines and books are full of project ideas using "found" materials.
Here are some crafts that use old egg cartons, either paper or Styrofoam (I like paper best, but either kind will do):
Creatures and critters: Using one or more sections from an egg carton and some pipe cleaners, you can make a whole zoo. If you paint the sections black and suspend them from strings, these make great spiders for Halloween. Add cotton balls dyed yellow to a painted yellow carton section to make an adorable Easter chick.
Similar to egg carton creatures, you can cut the sections apart and make egg carton flowers, using the carton pieces as the actual flower or the center of a larger flower and adding petals.
Egg cartons make good holders for crafts in progress (for drying painted Easter eggs, for example).
Egg cartons (or any other sectioned container) are great for sorting things like beads or any other small parts.
Fill each well of an egg carton with different colored paints for small projects.
Start seeds in each well. When they've sprouted, you can cut the sections apart and plant them (if they're the cardboard kind).
Put flowers you want to dry in silica gel (one in each section) and cover with the gel.
There are several strategy games that use sectioned game boards, which can be made out of egg cartons.
Styrofoam Trays, Cups, and Peanuts
Here are some crafts you can make out of Styrofoam:
Use a tray as a paint palette. Cut out a hole for your fingers.
An adult or older child can cut tray pieces out for stencils.
Rubber stamps can be expensive, but you can make your own out of Styrofoam meat trays. Clean the tray, cut out the shape, and glue it to an empty film canister, cork, or the end of a large eraser. Press into an inked stamp pad and you're ready to stamp.
Larger trays from bulk foods make good frames for artwork. You can also decorate them and use them as special trays.
Cut pieces of Styrofoam into different shapes and decorate to make coasters.
Decorate Styrofoam plates and use under plants to catch excess water.
Use leftover Styrofoam cups to grow seedlings (after you've used an egg carton to start them).
Styrofoam cups also work well as paint containers (just rinse them first) and homes for seedlings. Peanuts can be painted and strung together as "beads" or glued on as decorations.